Question: Let $f(x) = (x+2)^2-5$. If the domain of $f$ is all real numbers, then $f$ does not have an inverse function, but if we restrict the domain of $f$ to an interval $[c,\infty)$, then $f$ may have an inverse function. What is the smallest value of $c$ we can use here, so that $f$ does have an inverse function?
Solution: For $f$ to have an inverse function, it must not take any repeated value -- that is, we must not have $f(x_1)=f(x_2)$ for distinct $x_1$ and $x_2$ in its domain.

The graph of $y=(x+2)^2-5$ is a parabola with vertex at $(-2,-5)$:

[asy]
unitsize(0.2 cm);
Label f;

f.p=fontsize(4);

xaxis(-6,3,Ticks(f, 1.0, Size=1));
yaxis(-6,5,Ticks(f, 1.0, Size=1));

real g(real x)

{

return (x+2)^2-5;
}

draw(graph(g,-5.2,1.2));
dot((-2,-5));
label("Vertex: $(-2,-5)$", (-2,-5), SW);
[/asy] The axis of symmetry is the line $x=-2$, so for every $x$ less than $-2$, there is a corresponding $x$ greater than $-2$ where $f$ takes the same value. If we restrict the domain of $f$ to $[-2,\infty)$, then $f$ has no repeated values, as $f$ is increasing throughout its domain. But if we restrict the domain to $[c,\infty)$ where $c<-2$, then $f$ has repeated values. So, the smallest $c$ which will work is $c=\boxed{-2}$.